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I've seen the following definition of limits in the given books:

Laczkovich/Sós: Real Analysis

Let $f$ be defined on an open interval containing $a$, excluding perhaps $a$ itself. The limit of $f$ at $a$ exists and has value $b$ if for all $\epsilon>0$, there exists a $\delta >0$ such that

$$|f(x)-b|<\epsilon, \text{ if } 0<|x-a|<\delta$$

On the other hand:

Zorich: Mathematical Analysis 1

On a function $f:E\mapsto \Bbb{R}$.

$$\forall \epsilon>0\;\; \exists \delta >0 \;\; \forall x\in E \;\; (0<|x-a|<\delta \implies |f(x)-A|<\epsilon)$$

And this sides up with Spivak's definition on his Calculus and many other books I've seen. In the second definition, the inclusion of $\forall x\in E$ seems to tell me something very different. I tried to prove the inexistence of a limit in the Dirichlet function yersterday with the first definition and couldn't but when I read Zorich's definition, it was pretty straightforward so, are both definitions actually equivalent? I guess in the first definition, the authors expect us to understand $\forall x \in E$ by pointing to a figure in the page perhaps.

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    $\begingroup$ The set $E$ in the second definition is the open interval in the first definition. The point is that $f(x)$ only makes sense if $x$ is in the domain of $f$. In your first definition, this is implicit; in the second definition, it is made explicit. $\endgroup$ – Xander Henderson Mar 4 '19 at 0:28
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    $\begingroup$ ^^ This is better than either current answer, IMHO. $\endgroup$ – The Count Mar 4 '19 at 0:53
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They are exactly the same definition. In the second the domain $E$ is made explicit, while in the first it's not. Think about where are the $x$ coming from in the first definition.

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They are the same definition, with $b=A$, and $E$ being the open interval as defined above.

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